Question

The differential equation which is satisfied by all the curves, y = Ae^2x + Be^–x/2 , where A and B are non-zero constants, is-

• A. 2$\frac{d^{2}y}{dx^{2}}$ – 3$\frac{dy}{dx}$ – 2y = 0
• B. 2$\frac{d^{2}y}{dx^{2}}$ + 3$\frac{dy}{dx}$ – 2y = 0
• C. 2$\frac{d^{2}y}{dx^{2}}$ + 3$\frac{dy}{dx}$ + 2y = 0
• D. None of these

Answer 1

Answer: (A)

2$\frac{d^{2}y}{dx^{2}}$ – 3$\frac{dy}{dx}$ – 2y = 0

Answer 2

We have, y = Ae^2x + Be^–x/2 ... (1)

Ž$\frac{dy}{dx}$ = 2Ae^2x – $\frac{1}{2}$Be^–x/2 ... (2)

Ž$\frac{d^{2}y}{dx^{2}}$ = 4Ae^2x + $\frac{1}{4}$Be^–x/2 ... (3)

Eliminating A between equations (1) and (2), we get

Be^–x/2= $\frac{2}{5}$ $\left( 2y - \frac{dy}{dx} \right)$ ... (4)

\ From (1), Ae^2x = y – Be^–x/2

= y – $\frac { 2 } { 5 }$ $\left( 2y - \frac{dy}{dx} \right)$

ŽAe^2x= $\frac { 1 } { 5 }$ $\left( y + 2\frac{dy}{dx} \right)$ ... (5)

So, from equations (3), (4) and (5), we get

$\frac{d^{2}y}{dx^{2}}$ = $\frac { 2 } { 5 }$ $\left( 2y - \frac{dy}{dx} \right)$

Ž 2 $\frac{d^{2}y}{dx^{2}}$ – 3 $\frac{dy}{dx}$ – 2y = 0